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Generalized resolvents and spectrum for a certain class of perturbed symmetric operators. (English) Zbl 1091.47006

This paper is concerned with the study of spectral properties for a certain class of linear symmetric operators \(T\), defined in the Hilbert space \(H\) and of the form \(T=A+B\), where \(A\) is a closed linear symmetric operator, with nondensely defined domain in general, \(D(A)\subset H\), and \(B\) is a finite-rank operator of the form \(Bf=\sum\limits^n_{k=1}a_k(f,y_k) y_k\), where \(\{y_k, 1\leq k\leq n\}\subset H\) is a linearly independent system, \(\{a_k,1\leq k\leq n\}\subset \mathbb{R}\). For such a class of operators \(T\) with equal and finite deficiency indices, the author gives a classification of the spectrum using the Weinstein-Aronszajn formula, and obtains an expression of the generalized resolvent.

MSC:

47A10 Spectrum, resolvent
47A55 Perturbation theory of linear operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
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